Jmat.Real.erfi, branch x greater than or equal to 5

Average Accuracy: 95.6% → 98.1%

Time: 16.3s

Precision: binary64

Cost: 53056

\[x \geq 0.5\]

Math

\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

↓

\[\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{x \cdot x}\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
  (+
   (+
    (+
     (/ 1.0 (fabs x))
     (*
      (/ 1.0 2.0)
      (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))
    (*
     (/ 3.0 4.0)
     (*
      (*
       (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))
       (/ 1.0 (fabs x)))
      (/ 1.0 (fabs x)))))
   (*
    (/ 15.0 8.0)
    (*
     (*
      (*
       (*
        (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))
        (/ 1.0 (fabs x)))
       (/ 1.0 (fabs x)))
      (/ 1.0 (fabs x)))
     (/ 1.0 (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ 1.0 PI))
  (fma
   (+ (/ 1.0 x) (/ 0.75 (pow x 5.0)))
   (pow (exp x) x)
   (* (+ (/ 0.5 x) (/ 1.875 (pow x 5.0))) (/ (exp (* x x)) (* x x))))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * ((((1.0 / fabs(x)) + ((1.0 / 2.0) * (((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))))) + ((3.0 / 4.0) * (((((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))))) + ((15.0 / 8.0) * (((((((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x)))));
}

↓

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * fma(((1.0 / x) + (0.75 / pow(x, 5.0))), pow(exp(x), x), (((0.5 / x) + (1.875 / pow(x, 5.0))) * (exp((x * x)) / (x * x))));
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(Float64(1.0 / abs(x)) + Float64(Float64(1.0 / 2.0) * Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))) + Float64(Float64(3.0 / 4.0) * Float64(Float64(Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))) + Float64(Float64(15.0 / 8.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))))
end

↓

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(Float64(1.0 / x) + Float64(0.75 / (x ^ 5.0))), (exp(x) ^ x), Float64(Float64(Float64(0.5 / x) + Float64(1.875 / (x ^ 5.0))) * Float64(exp(Float64(x * x)) / Float64(x * x)))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] + N[(N[(N[(0.5 / x), $MachinePrecision] + N[(1.875 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} \]
   Proof

   [Start] 95.6

   \[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   associate-+l+ [=>] 95.6

   \[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Taylor expanded in x around inf 95.8%

   \[\leadsto \color{blue}{\left(e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \frac{e^{{x}^{2}} \cdot \left(0.5 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}} \]

4. Simplified 98.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{x \cdot x}\right)} \]
   Proof

   [Start] 95.8	\[ \left(e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \frac{e^{{x}^{2}} \cdot \left(0.5 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}} \]
   distribute-rgt-out [=>] 95.8	\[ \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{e^{{x}^{2}} \cdot \left(0.5 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)}{{x}^{2}}\right)} \]
   *-commutative [=>] 95.8	\[ \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) \cdot e^{{x}^{2}}} + \frac{e^{{x}^{2}} \cdot \left(0.5 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)}{{x}^{2}}\right) \]
   unpow2 [=>] 95.8	\[ \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) \cdot e^{\color{blue}{x \cdot x}} + \frac{e^{{x}^{2}} \cdot \left(0.5 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)}{{x}^{2}}\right) \]
   fma-def [=>] 95.8	\[ \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}, e^{x \cdot x}, \frac{e^{{x}^{2}} \cdot \left(0.5 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)}{{x}^{2}}\right)} \]

5. Taylor expanded in x around inf 98.1%

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \color{blue}{\frac{e^{{x}^{2}}}{{x}^{2}}}\right) \]

6. Simplified 98.1%

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{x \cdot x}}\right) \]
   Proof

   [Start] 98.1	\[ \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \frac{e^{{x}^{2}}}{{x}^{2}}\right) \]
   unpow2 [=>] 98.1	\[ \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \frac{e^{\color{blue}{x \cdot x}}}{{x}^{2}}\right) \]
   unpow2 [=>] 98.1	\[ \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\color{blue}{x \cdot x}}\right) \]

7. Final simplification 98.1%

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}, {\left(e^{x}\right)}^{x}, \left(\frac{0.5}{x} + \frac{1.875}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{x \cdot x}\right) \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	33600

\[\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot \left({\pi}^{-0.5} \cdot \frac{{\left(e^{x}\right)}^{x}}{x}\right) \]

Alternative 2
Accuracy	98.1%
Cost	33600

\[\left({\left(e^{x}\right)}^{x} \cdot \frac{{\pi}^{-0.5}}{x}\right) \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

Alternative 3
Accuracy	98.0%
Cost	33536

\[\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{{\left(e^{x}\right)}^{x}}{x}}{\sqrt{\pi}} \]

Alternative 4
Accuracy	95.8%
Cost	27264

\[\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{{\pi}^{-0.5}}{\frac{x}{e^{x \cdot x}}} \]

Alternative 5
Accuracy	95.8%
Cost	27200

\[\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{x}}{\sqrt{\pi}} \]

Alternative 6
Accuracy	30.3%
Cost	26560

\[\sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right) \]

Alternative 7
Accuracy	24.7%
Cost	26048

\[\sqrt{\frac{1}{\pi}} \cdot \frac{{\left(e^{x}\right)}^{x}}{x} \]

Alternative 8
Accuracy	24.7%
Cost	19712

\[\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{x} \]

Alternative 9
Accuracy	11.2%
Cost	19648

\[1.875 \cdot \frac{\sqrt{\frac{1}{\pi}}}{{x}^{7}} \]

Alternative 10
Accuracy	10.8%
Cost	19520

\[\sqrt{0.5625 \cdot \frac{{x}^{-10}}{\pi}} \]

Reproduce

herbie shell --seed 2023136 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))